Lemma 109.7.1. There exists a local ring $R$ and a maximal ideal $\mathfrak m$ such that the completion $R^\wedge$ of $R$ with respect to $\mathfrak m$ has the following properties

1. $R^\wedge$ is local, but its maximal ideal is not equal to $\mathfrak m R^\wedge$,

2. $R^\wedge$ is not a complete local ring, and

3. $R^\wedge$ is not $\mathfrak m$-adically complete as an $R$-module.

Proof. This follows from the discussion above as (with $R = k[x_1, x_2, x_3, \ldots ]$) the completion of the localization $R_{\mathfrak m}$ is equal to the completion of $R$. $\square$

There are also:

• 6 comment(s) on Section 109.7: Noncomplete completion

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05JC. Beware of the difference between the letter 'O' and the digit '0'.